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Properties

Label 2-3e3-27.14-c6-0-7
Degree 22
Conductor 2727
Sign 0.596−0.802i0.596 - 0.802i
Analytic cond. 6.211466.21146
Root an. cond. 2.492282.49228
Motivic weight 66
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.97 + 1.75i)2-s + (24.1 + 12.1i)3-s + (36.1 − 13.1i)4-s + (60.9 − 72.6i)5-s + (−261. − 78.3i)6-s + (−133. − 48.6i)7-s + (223. − 129. i)8-s + (435. + 584. i)9-s + (−480. + 832. i)10-s + (1.62e3 + 1.93e3i)11-s + (1.03e3 + 120. i)12-s + (−126. + 717. i)13-s + (1.41e3 + 250. i)14-s + (2.35e3 − 1.01e3i)15-s + (−3.88e3 + 3.26e3i)16-s + (4.21e3 + 2.43e3i)17-s + ⋯
L(s)  = 1  + (−1.24 + 0.219i)2-s + (0.893 + 0.448i)3-s + (0.565 − 0.205i)4-s + (0.487 − 0.581i)5-s + (−1.21 − 0.362i)6-s + (−0.390 − 0.141i)7-s + (0.436 − 0.252i)8-s + (0.597 + 0.801i)9-s + (−0.480 + 0.832i)10-s + (1.22 + 1.45i)11-s + (0.597 + 0.0696i)12-s + (−0.0575 + 0.326i)13-s + (0.517 + 0.0912i)14-s + (0.696 − 0.300i)15-s + (−0.949 + 0.796i)16-s + (0.856 + 0.494i)17-s + ⋯

Functional equation

Λ(s)=(27s/2 ΓC(s) L(s)=((0.596−0.802i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(27s/2 ΓC(s+3) L(s)=((0.596−0.802i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2727    =    333^{3}
Sign: 0.596−0.802i0.596 - 0.802i
Analytic conductor: 6.211466.21146
Root analytic conductor: 2.492282.49228
Motivic weight: 66
Rational: no
Arithmetic: yes
Character: χ27(14,⋅)\chi_{27} (14, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 27, ( :3), 0.596−0.802i)(2,\ 27,\ (\ :3),\ 0.596 - 0.802i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 1.08295+0.544511i1.08295 + 0.544511i
L(12)L(\frac12) ≈\approx 1.08295+0.544511i1.08295 + 0.544511i
L(4)L(4) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(−24.1−12.1i)T 1 + (-24.1 - 12.1i)T
good2 1+(9.97−1.75i)T+(60.1−21.8i)T2 1 + (9.97 - 1.75i)T + (60.1 - 21.8i)T^{2}
5 1+(−60.9+72.6i)T+(−2.71e3−1.53e4i)T2 1 + (-60.9 + 72.6i)T + (-2.71e3 - 1.53e4i)T^{2}
7 1+(133.+48.6i)T+(9.01e4+7.56e4i)T2 1 + (133. + 48.6i)T + (9.01e4 + 7.56e4i)T^{2}
11 1+(−1.62e3−1.93e3i)T+(−3.07e5+1.74e6i)T2 1 + (-1.62e3 - 1.93e3i)T + (-3.07e5 + 1.74e6i)T^{2}
13 1+(126.−717.i)T+(−4.53e6−1.65e6i)T2 1 + (126. - 717. i)T + (-4.53e6 - 1.65e6i)T^{2}
17 1+(−4.21e3−2.43e3i)T+(1.20e7+2.09e7i)T2 1 + (-4.21e3 - 2.43e3i)T + (1.20e7 + 2.09e7i)T^{2}
19 1+(−62.0−107.i)T+(−2.35e7+4.07e7i)T2 1 + (-62.0 - 107. i)T + (-2.35e7 + 4.07e7i)T^{2}
23 1+(4.13e3+1.13e4i)T+(−1.13e8+9.51e7i)T2 1 + (4.13e3 + 1.13e4i)T + (-1.13e8 + 9.51e7i)T^{2}
29 1+(−2.21e4+3.91e3i)T+(5.58e8−2.03e8i)T2 1 + (-2.21e4 + 3.91e3i)T + (5.58e8 - 2.03e8i)T^{2}
31 1+(−1.41e4+5.13e3i)T+(6.79e8−5.70e8i)T2 1 + (-1.41e4 + 5.13e3i)T + (6.79e8 - 5.70e8i)T^{2}
37 1+(−1.72e4+2.99e4i)T+(−1.28e9−2.22e9i)T2 1 + (-1.72e4 + 2.99e4i)T + (-1.28e9 - 2.22e9i)T^{2}
41 1+(8.70e4+1.53e4i)T+(4.46e9+1.62e9i)T2 1 + (8.70e4 + 1.53e4i)T + (4.46e9 + 1.62e9i)T^{2}
43 1+(4.95e3−4.15e3i)T+(1.09e9−6.22e9i)T2 1 + (4.95e3 - 4.15e3i)T + (1.09e9 - 6.22e9i)T^{2}
47 1+(−5.96e4+1.63e5i)T+(−8.25e9−6.92e9i)T2 1 + (-5.96e4 + 1.63e5i)T + (-8.25e9 - 6.92e9i)T^{2}
53 1−2.77e5iT−2.21e10T2 1 - 2.77e5iT - 2.21e10T^{2}
59 1+(−6.67e4+7.95e4i)T+(−7.32e9−4.15e10i)T2 1 + (-6.67e4 + 7.95e4i)T + (-7.32e9 - 4.15e10i)T^{2}
61 1+(2.84e5+1.03e5i)T+(3.94e10+3.31e10i)T2 1 + (2.84e5 + 1.03e5i)T + (3.94e10 + 3.31e10i)T^{2}
67 1+(7.05e3−3.99e4i)T+(−8.50e10−3.09e10i)T2 1 + (7.05e3 - 3.99e4i)T + (-8.50e10 - 3.09e10i)T^{2}
71 1+(8.73e3+5.04e3i)T+(6.40e10+1.10e11i)T2 1 + (8.73e3 + 5.04e3i)T + (6.40e10 + 1.10e11i)T^{2}
73 1+(−1.59e5−2.75e5i)T+(−7.56e10+1.31e11i)T2 1 + (-1.59e5 - 2.75e5i)T + (-7.56e10 + 1.31e11i)T^{2}
79 1+(−1.17e4−6.63e4i)T+(−2.28e11+8.31e10i)T2 1 + (-1.17e4 - 6.63e4i)T + (-2.28e11 + 8.31e10i)T^{2}
83 1+(9.75e5−1.72e5i)T+(3.07e11−1.11e11i)T2 1 + (9.75e5 - 1.72e5i)T + (3.07e11 - 1.11e11i)T^{2}
89 1+(1.69e5−9.79e4i)T+(2.48e11−4.30e11i)T2 1 + (1.69e5 - 9.79e4i)T + (2.48e11 - 4.30e11i)T^{2}
97 1+(−9.81e5+8.23e5i)T+(1.44e11−8.20e11i)T2 1 + (-9.81e5 + 8.23e5i)T + (1.44e11 - 8.20e11i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−16.59143083879334815328348446552, −15.18990271789207063995756824233, −13.90011252943384485070535928477, −12.48695764975568683232251265263, −10.15671272888729288920391646092, −9.513563306340111874200109526233, −8.487845488228712791259574822730, −7.00903951116024956798215645228, −4.30239394815521649551556003621, −1.61269623752675205124917179934, 1.13650830990174735426884987872, 3.06567213644819312787242656600, 6.46417044341946819333415029701, 8.036402912990855935773208761640, 9.161037278793227408384916949424, 10.14009661499303150013821996697, 11.74726504697007505399474797805, 13.65139610327940066077269834879, 14.35079063271483588461751883541, 16.12586247732643564364489537441

Graph of the ZZ-function along the critical line